6. A bar 100 cm long, with insulated sides, has its ends kept at 0°C and 100°C until steady state conditions prevail. The two ends are then suddenly insulated and kept so. Find the temperature distribution.
Answers
Answer:
The heat equation is
Let u = X(x) . T(t) be the solution of (1), where „X‟ is a function of „x‟ alone and „T‟ is a function of „t‟ alone.
Substituting these in (1), we get
Now the left side of (2) is a function of „x‟ alone and the right side is a function of „t‟ alone. Since „x‟ and „t‟ are independent variables, (2) can be true only if each side is equal to a constant.
Hence, we get X′′ - kX = 0 and T′ -a2kT=0.-------------- (3).
Solving equations (3), we get
(i) when „k‟, is say positive and k = l2
X = c1 elx + c2 e - lx
(iii) when „k‟ is zero.
X = c7 x + c8
T = c9
Thus the various possible solutions of the heat equation (1) are
Of these three solutions, we have to choose that solution which suits the physical nature of the problem and the given boundary conditions. As we are dealing with problems on heat flow, u(x,t) must be a transient solution such that „u‟ is to decrease with the increase of time „t‟.
Therefore, the solution given by (5),
is the only suitable solution of the heat equation.